A Karhunen–Loève expansion for a mean-centered Brownian bridge

The processes of the form YK(t)=B(t)-6Kt(1-t)∫01B(u)du, where K   is a constant, and B(·)B(·) a Brownian bridge, are investigated. We show that Y0(·)Y0(·) and Y2(·)Y2(·) are both Brownian bridges, and establish the independence of Y1(·)Y1(·) and ∫01B(u)du, this implying that the law of Y1(·)Y1(·) coincides with the conditional law of B  , given that ∫01B(u)du=0. We provide the Karhunen–Loève expansion on [0,1][0,1] of Y1(·)Y1(·), making use of the Bessel functions J1/2J1/2 and J3/2J3/2. Applications and variants of these results are discussed. In particular, we establish a comparison theorem concerning the supremum distributions of YK′(·)YK′(·) and YK″(·)YK″(·) on [0,1][0,1].